investigation to define the propagation … to define the propagation ... investigation to define...
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t
~ N A S A C O N T R A C T O R N A S A C R - 7 3 6 1 - 1 ._
R E P O R T IC
I f.
40 m
INVESTIGATION TO DEFINE THE PROPAGATION CHARACTERISTICS OF A FINITE AMPLITUDE ACOUSTIC PRESSURE WAVE
b&A. C. Peter mzd J . W. Cottrell
r"r e p i e u ' &y
NORTH AMERICAN AVIATION, INC. '2
Downey, Calif. 2 for George C. itinrshrzll Space Flight Center
N A T I O N A L A E R O N A U T I C S A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. A P R I L 1 9 6 7
https://ntrs.nasa.gov/search.jsp?R=19670012652 2018-06-24T04:55:45+00:00Z
NASA CR-736
INVESTIGATION T O DEFINE THE PROPAGATION CHARACTERISTICS
O F A FINITE AMPLITUDE ACOUSTIC PRESSURE WAVE
By A. C . Peter and J. W. C o t t r e l l
Dis t r ibu t ion of t h i s report is provided i n t h e i n t e r e s t of i n f o r m a t i o n exchange . Responsibi l i ty for the con ten t s resides i n the au tho r or organizat ion tha t p r e p a r e d it.
,,6 P r e p a r e d u n d e r C o n t r a c t No. I NAS 8 - 1 1 4 4 f b y
NORTH AMERICAN AVIATION, INC. Downey , Calif .
for G e o r g e C . M a r s h a l l Space F l ight C e n t e r
N A T I O N A L AERONAUT ICs AND SPACE ADMINISTRATION
For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00
r ! I r; ,? i , : P i s . ( 7 3 ) ::<. ' a r 9 i L ' \ q; - - f 1 < / ' j
J V C"REWQRD
This report presents work performed hy North
American Aviation, Inc. , Space a.nd Informatton
Systems Division in fulfillment of Contract
NAS8-11441, '!lnve/sLLgation t o Dedine t h e Ptropaga;tion
ChanactehinLLcn 05 a Finite AwpUude Paenauae Wave."
This study has been administered by the Unsteady
Aerodynamics Branch, National Aeronautics and Space
Administration, George C . Marshall Space F l i g h t
Center, Huntsville, Alabama.
Dr. F. C . Hung acted as Program Manager for
North American Aviation, Inc. Mr. A. C. Peter and
J . M. Cottrell were the investigators on the project.
iii
ACKNONLEDGEMENTS
The P r i n c i p a l I n v e s t i g a t o r would l i k e t o
express h i s thanks t o D r . T. C. L i f o r h i s h e l p
on var ious p a r t s o f t he p r o j e c t and cons tan t
encouragement throughout.
express h i s thanks t o Prof . J . Laufer f o r h i s
c r i t i c a l comments concerning c e r t a i n p a r t s o f
t h i s research p r o j e c t , t o D r . R. Kaplan f o r h i s
i n t e r e s t and t ime taken up by discussion, t o
R . H a l l i b u r t o n and D r . E . T. Benedikt f o r t h e i r
k i n d encouragement and suggestions.
He would a l s o l i k e t o
iv
.
SUMMARY
A t h e o r e t i c a l a n a l y s i s o f the propagat ion c h a r a c t e r i s t i c s o f a f i n i t e
amp1 i tude pressure wave i s presented i n t h e f o l 1 owing sect ions.
The a n a l y s i s at tempts t o study the c o n t r i b u t i o n o f entropy-producing
It r e s u l t s i n a reg ions t o t h e mechanism o f aerodynamic noise generat ion.
n o n - l i n e a r convec t ive wave equat ion i n terms o f ent ropy and a thermodynamic
J f u n c t i o n .
those used i n c l a s s i c a l l i t e r a t u r e i s obtained. An i d e a l i z a t i o n of the
processes considered permi ts t h e uncoupl ing o f t h e equat ions o f mot ion
w i t h a consequent c o n s t r u c t i o n o f an acoust ic analogy t r e a t i n g shock wave
emission o f f i n i t e ampl i tude acous t ic waves.
A d i r e c t analogy between t h e der ived governing equat ion and
An engineer ing approach of t h i s analogy i s r e f l e c t e d i n t h e concept
o f an extended p l u g nozz le whose f u n c t i o n i t i s t o f a c i l i t a t e aerodynamic
n o i s e a t t e n u a t i o n by modify ing t h e entropy-producing reg ions.
V
TABLE OF CONTENTS
Section Page
I . B A S I C THEORY
1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . 1
2 . THE NON-ISENTROPIC CONVECTED WAVE EQUATION . . . . . . 2
3 . THE THERMODYNAMIC VARIABLE J . . . . . . . . . . . . . 10
4 . LUMl'AK13UIY w i i n fhl3111lU I I I L ~ ~ \ ~ ~ ~ . . . . . . . . . . 18
5 . SOME A D D I T I O N A L ASPECTS OF THE GOVERNING EQUATION . . 24
6 . THE PROCESS J . CONSTANT . . . . . . . . . . . . . . . 27
7 . THE CASE OF MODERATE SHOCK WAVES . . . . . . . . . . . 34
............. I T T I I r v l r - r T m l r TUcnDTcC
I 1 . ENGINEERING A P P L I C A T I O N
8 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . 41
9 . ENGINEERING REPRESENTAT ION . . . . . . . . . . . . . . 42
10 . AN ACOUSTIC ANP-LOGY . . . . . . . . . . . . . . . . . 48
11 . THE CONICAL SHOCK LAYER . . . . . . . . . . . . . . . 58
12 . THE TRIGGERING MECHANISM . . . . . . . . . . . . . . 71
13 . EXPERIMENTAL V E R I F I C A T I O N . . . . . . . . . . . . . . 7 6
CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . 87
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . 89
vii
FIGURES
Page Number
Figure Number
1
2
3
4
5
6a
6b
7a
7b
Pressure vs. Densi ty f o r J=Constant Process Appl ied
t o Shock Layer T r a n s i t i o n Flow . . . . . . . . . . . . . The Non- Isentrop ic Propagat ion Speed a vs Entropy
Appl ied t o Shock Layer T r a n s i t i o n Flow . . . . . . . . . The Parameter B /ao vs Upstream Mach Number f o r J=Constant
Process Appl ied t o a Normal Shock T r a n s i t i o n Flow . . . 38
Pressure vs. Entropy f o r J=Constant Process Appl ied t o
28
32 2
Shock Layer T r a n s i t i o n Flow . . . . . . . . . . . . . 39
Densi ty vs Entropy f o r J=Constant Process Appl ied t o
Shock Layer T r a n s i t i o n Flow . . . . . . . . . . . . . 40
Po la r Graphs o f Lines o f Constant I n t e n s i t y i n the Noise
F i e l d o f a 25" Conical Shock Showing the Frequency
Dependent D i r e c t i o n a l i t y . . . . . . . . . . . . . . . 63
Distance From Shock Versus Spher ica l Angle f o r L ines o f
Constant I n t e n s i t y i n t h e F i e l d of a 25' Conical Shock . Polar Graphs of L ines of Constant I n t e n s i t y i n the Noise
F i e l d o f a 45" Conical Shock Showing the Frequency
Dependent D i r e c t i o n a l i t y . . . . . . . . . . . . . . . 65
Distance from Shock versus Spher ica l Angle f o r L ines of
Constant I n t e n s i t y i n t h e F i e l d o f a 45" Conical Shock .
64
66
viil
F i gure Number
8a
8b
8c
0-l OU
9
10
1 l a
l l b
l l c
12
D i r e c t i o n a l i t y F i e l d o f a Parabalo ida l Shock a t
h = u/a0 = .lO(cgs u n i t s )
D i r e c t i o n a l i t y F i e l d o f a Parabalo ida l Shock a t
. . . . . . . . . . . . . . .
'b' /= w / a 0 = .50 (cgs u n i t s ) . . . . . . . . . . . . . . . D i r e c t i o n a l i t y F i e l d of a Parabaio ida i Shock a t
h = u/a0 = 1.00 (cgs u n i t s ) . . . . . . . . . . . . . . D i r e c t i z n a ! i t y F i e l d nf 5 Parahalo ida l Shock a t
h = d u o
A C h a r a c t e r i s t i c CRT P l o t of Spectrum f o r Two Typ ica l Time
I n p u t Funct ions (Simular ion o f Smith 's Experiment; Y = 50"
Comparison o f Frequency D i r e c t i v i t y I n d i c e s as Computed by
Model and as Determined Exper imenta l ly by Smith
Comparison of Spectra Computed by Model w i t h Those Obtained
= 2.00 (cgs u n i t s ) . . . . . . . . . . . . . .
. . . .
Exper imenta l l y by Smith (Program F i r i n g No. 1 )
Exper imenta l l y by Smith (Program F i r i n g No. 1 )
Exper imenta l l y by Smith (Program F i r i n g No. 1 )
(Program F i r i n g No. 1) . . . . . . . . . . . . . . . . .
. . . . Comparison of Spectra Computed by Model w i t h Those Obtained
. . . . Comparison of Spectra Computed by Model w i t h Those Obtained
. . . . O v e r a l l D i r e c t i v i t y S imu la t ion o f Smi th 's Experiments
Page Number
67
68
69
70
75
80
81
82
83
84
ix
I. BASIC THEORY
1. INTRODUCTION
This t h e o r e t i c a l ana lys i s o f t h e propagation c h a r a c t e r i s t i c s o f a
f i n i t e ampl i tude pressure wave has been concerned w i t h the c o n t r i b u t i o n o f
entropy-producing reg ions t o the mechanism o f aerodynamic noise generat ion
approach
ons on
i n t h e exhaust o f a supersonic nozzle. It i s hoped t h a t t he presen
w i l l tend t o supplement some aspects o f t he contemporary i n v e s t i g a t
t he s u b j e c t o f aerodynamic noise generation.
A f i r s t at tempt t o i n v e s t i g a t e t h e f e a s i b i l i t y o f aerodynamic no ise
a t t e n u a t i o n by modify ing t h e geometry of entropy-produci ng regions ( i n t h i s
p a r t i c u l a r case, shock waves) r e s u l t e d i n the concept o f an extended p l u g
nozz le (Pe te r and Kamo, 1963).
e a r l y ideas on a more r i go rous bas is by consider ing non-- isentropic processes
coupled w i t h t h e equat ions o f motion o f f l u i d dynamics.
The present analys is at tempts t o p u t these
The a n a l y s i s r e s u l t s i n a non l i nea r convect ive wave equat ion i n terms o f
entropy and a thermodynamic J - func t i on .
speed C( i s de f i ned and expressed i n terms o f t he ent ropy funct ion. This
propagat ion speed reduces t o the i s e n t r o p i c speed o f sound as the ent ropy
p roduc t i on i n the process tends t o zero. A proper choice o f the thermo-
dynamic J - f u n c t i o n a l lows a d i r e c t analogy between the de r i ved governing
equat ion and those used i n c l a s s i c a l l i t e r a t u r e ( L i g h t h i l l , 1952, P h i l l i p s ,
1960, Ribner, 1954-62) when the entropy terms a re assumed small .
A non- isent rop ic propagat ion
The
i z i n g the process equat ions o f mot on a re s i m p l i f i e d and uncoupled by idea
-1-
t o c o n s i s t o f s t r a i g h t l i n e s i n the I-S plane, each r e f e r r i n g t o a
d i f f e r e n t reg ion i n space. Th is i d e a l i z a t i o n i m p l i e s a f a r f i e l d no ise
analys is s ince near f i e l d e f f e c t s admit of mixed J-S processes t a k i n g
place.
It i s a l s o shown t h a t , w i t h i n t h e framework o f t h i s ana lys is , t h e
pressure, dens i ty and temperature f u n c t i o n s depend on the h i s t o r y o f t h e
process r a t h e r than t h e instantaneous s t a t e s o f the f l u i d p a r t i c l e s . I t
appears, a1 so, t h a t , whereas a1 1 f u n c t i o n a l values o f the thermodynamic
var iab les (e.g., p, p , T , e t c . ) a re cont inuous on the boundary o f a non-
i s e n t r o p i c reg ion ( a d j o i n i n g an i s e n t r o p i c r e g i o n ) t h e i r d e r i v a t i v e s are no t .
The non- isen t rop ic propagat ion speed a d i f f e r s f rom i t s i s e n t r o p i c counter-
p a r t , a t the end o f t h e non- isen t rop ic process, by a f a c t o r which i s c lose,
bu t n o t equal to , u n i t y .
entropy produced d u r i n g the process and tends t o u n i t y as the ent ropy pro-
duct ion tends t o zero. Moreover, i t a l s o appears t h a t , even though t h e
entropy v a r i a t i o n f o r weak shocks i s o f a t h i r d o rder o f magnitude, the
e f f e c t s o f entropy produc t ion cannot be d isregarded s ince, i n the governing
d i f f e r e n t i a l equation, ent ropy e f f e c t s are o f a f i r s t - o r d e r magnitude due
t o an a d d i t i o n a l f a c t o r appear ing i n t h e ana lys is .
Th is f a c t o r depends d i r e c t l y upon t h e amount o f
? . THE NON-ISENTROPIC CONVECTED WAVE EQUATION
I n t h e proceeding a n a l y s i s i t i s assumed t h a t a p e r f e c t gas s a t i s f y i n g
the equat ion o f s t a t e
p = pRT ( 2 . 1 I
forms a r e g i o n of ent ropy product ion.
t h e f o l l o w i n g equat ions of mot ion descr ibe t h e f l o w c h a r a c t e r i s t i c s
I n a f i x e d Car tes ian Reference Frame
-2-
r 1
( 2 . 3 )
(2.41
The f o i i o w i n g n o t a t f i x is used:
P
P T
= dens i ty of t h e f l u i d
= pressure o f t he f l u i d
= temperature o f t he f l u i d
= v e l o c i t y v e c t o r i U
s = entropy
u = v i s c o s i t y c o e f f i c i e n t
h = heat t r a n s f e r c o e f f i c i e n t
= body forces vec to r Fi 2 = heat sources
R = gas constant
9 = d i s s i p a t i o n f u n c t i o n
= viscous p a r t o f t he s t r e s s tenso r i j T
I n terms of t h e above n o t a t i o n the fo l l ow ing s u b s i d i a r y r e l a t i o n s are needed:
-3-
wi th
= Kronecker d e l t a " j
Since the a n a l y s i s o f entropy-producing regions imp1 i e s a s t rong
coupl ing between the mechanical cond i t i ons .(momentum equat ion) and thermo-
dynamic cond i t i ons (energy equat ion) , i t w i l l be use fu l t o p u t the momentum
equat ion i n terms of thermodynamic va r iab les .
divergence o f the momentum Equat ion (2.3) t o o b t a i n
To do so we f i r s t take the
L e t J be a thermodynamic v a r i a b l e ( t o be de f i ned a t a l a t e r stage)
and consider the pressure t o be a f u n c t i o n o f d e n s i t y and the v a r i a b l e J.
Under these cond i t i ons the pressure d i f f e r e n t i a l may be w r i t t e n i n the form
-4-
We nex t d e f i n e a q u a n t i t y a having the dimensions o f v e l o c i t y and
g iven by
2 a = [?j3
Under these c o n d i t i o n s t h e pressure g rad ien t t e r m i n Equat ion ( 2 . 5 ) may be
w r i t t e n as
I n view o f r e l a t i o n s ( 2 . 2 ) and (2 .b ) , Equation ( 2 . 5 ) takes on the f o l l o w i n g
form
i axi ax 3 a l a j
axi p ax = - [ - - ( T ~ ~ ) i Fi - - - (2.91
Now l e t us cons ider the en t ropy func t ion . To s im-p l i f y the de r i va t i on ,
a c a l o r i c a l l y p e r f e c t gas w i l l be assumed t o y i e l d
(2.10)
where cy and c are the s p e c i f i c heats o f t he gas and y i s t h e i r r a t i o . P
A f u n c t i o n G w i l l now be de f ined i n such a manner t h a t
G =-cv Ln[b) (2.17)
Under these cond i t i ons , R e l a t i o n (2 .10 ) becomes s imp ly :
- 5-
' (S + GI P
C ( 2 . 1 2 )
I n us ing Re la t i on ( 2 . 1 2 ) i n Equation ( 2 . 9 ) t he thermodynamic v a r i a b l e s
S and J w i l l be regarded as independent, va r iab les . As mentioned p rev ious l y ,
the thermodynamic v a r i a b l e J w i l l be de f i ned l a t e r . Before us ing R e l a t i o n
( 2 . 1 2 ) i n t he d i f f e r e n t i a l equat ion ( 2 . 9 ) , i t w i l l be convenient t o no te t h e
form o f t he d e r i v a t i v e s o f Equat ion ( 2 . 1 2 ) . I n accordance w i t h the above
hypothesis we have
Hence, t a k i n g the f i r s t and second d i f f e r e n t i a l s o f Equat ion ( 2 . 1 2 ) , the
f o l l o w i n g r e l a t i o n s are ob ta ined
( 2 . 1 3 )
I ( 2 . 1 4 )
Consider nex t t he d i f f e r e n t i a l Equat ion ( 2 . 9 ) and w r i t e i t i n the
f o l l o w i n g form:
-6-
Fro% Equaticns ! 2 . 7 3 ) and ( 2 . 7 4 ) the l a s t equation may be written as
Collecting terms i n the above equation, the following relat ion i s
obtained:
-7-
( 2 . 1 6 )
To s i m p l i f y t he n o t a t i o n we in t roduce two func t i ons H and K given
by ;
and w r i t e the l a s t express ion o f Equat ion ( 2 . 1 6 ) i n t h e fo rm
aJ ( 2 . 1 8 )
I n view o f Equations ( 2 . 1 7 ) and ( 2 . 1 8 1 , Equat ion ( 2 . 1 6 ) becomes
- 8-
Simp l i f y i ng the above, the f o l l o w i n g d i f f e r e n t i a l equat ion i s obtained:
t
-9-
Equation ( 2 . 1 9 1 represents a nonlinear convective wave equation i n
terms of the two thermodynamic variables S and J. I t implies t h a t , in a
non-isentropic region, entropy changes are propagated convectively within
the region. The source dis t r ibut ion i s represented by the right-hand s ide
of Equat ion (2 .191 and i s shown t o depend upon the nature of the thermodynamic
variable J and also upon the l a s t two terms of the equation representing
the e f fec ts of viscosi ty , body forces and velocity fluctuations.
of the character is t ics i s d i rec t ly a function of the propagation speed
The shape
(2 .201
Formal ly , t h i s propagation speed depends d i rec t ly upon the entropy
s ta te of the f lu id par t ic les and also upon the process J = constant.
these two thermodynamic variables i t i s a function of the space variables
x and the time t. I n addition, the form of the equation indicates
dissipative processes due t o the appearance of f i r s t derivatives b o t h i n
space and time and a lso due t o the highly nonlinear character of the pro-
cesses involved.
since a l l coefficients on the l e f t of the above equation are functions of
entropy, i . e . , of the dependent variable (except the coeff ic ient of the
second time derivative which has been incorporated in the forcing function
on the right-hand side by su i tab le divis ion) .
Through
i
This i s reflected by the non-linearity o f the equation
3 . THE THERMODYNAMIC VARIABLE J
Equation ( 2 . 1 9 ) governing’ the flow in non-isentropic regions admits
t w o independent thermodynamic variables, namely, the entropy S and the
variable J which, so f a r , i s of an a rb i t ra ry character. The remaining
-10-
va r iab les , whose mathematical forms a re sought i n t h e present i n v e s t i g a t i o n ,
e.g., t he pressure p , the d e n s i t y P , etc . , a re f u n c t i o n s o f t he space
coordinates x i ( i = 7,2,3) and the t ime R , through these two independent
va r iab les . To o b t a i n an a n a l y t i c a l l y meaningful r e s u l t f rom the prev ious
d e r i v a t i o n , t h e thermodynamic v a r i a b l e J must be un ique ly determined. Now,
i t appears t h a t t he unique de terminat ion o f t h i s v a r i a b l e cannot be achieved
f rom p u r e l y mathematical o r thermodynamic considerat ions, s ince h i t h i n the
framework o f these two sciences t h e va r iab le J need n o t be s p e c i f i e d
t o o b t a i n the equat ion represented by Relat!ion
demonstrated by the process o f t h e previous ana y s i s r e s u l t i n g i n the ac tua l
d e r i v a t i o n o f t h e governing equat ion w i thout recourse t o any hypothesis con-
cern ing the cha rac te r of t he J func t i on .
2 . 7 9 ) . Th is i s amply
To s p e c i f y the fo rm o f the thermodynamic J - func t i on i t i s necessary
t o s c r u t i n i z e some phys i ca l aspects o f the present d e r i v a t i o n . Thus, i f
T-I = rl(S,J)
w r i t t e n i n t h e form:
i s a g iven thermodynamic var iab le , i t s d i f f e r e n t i a l may be
The c o e f f i c i e n t of dJ i n Equation (3. l l r ep resen ts an i s e n t r o p i c process
w i t h J vary ing , whereas t h a t o f dS r e f l e c t s entropy v a r i a t i o n s du r ing
a J = cons tan t process.
producing r e g i o n the c o n t r i b u t i o n o f t he J = cons tan t process must
predominate.
I t i s apparent t h a t i n the case o f an entropy-
The cho ice of t he dimensional u n i t s f o r the J f u n c t i o n i s comple te ly
a r b i t r a r y i n s o f a r as keeping the process J = cons tan t an i n v a r i a n t .
such a dimensional change may be e f f e c t i v e l y accomplished by a d j o i n i n g
For
-1 1-
t o the d i f f e r e n t i a l dT a r b i t r a r y factors , w i t h o u t a f f e c t i n g t h e J = constant
process. I n t h e present case when non- i sen t rop i c pressure v a r i a t i o n s are
considered, t he s imp les t form w i l l be obta ined by a s c r i b i n g t o t h e v a r i a b l e
J t he dimensions o f pressure.
Dimensional cons iderat ions i n d i c a t e t h a t the f u n c t i o n :
where B i s a constant parameter having the dimensions o f v e l o c i t y f u l f i l l s
t h i s requirement.* For the present, t he form o f t he J f unc t i on as given
i n Equation ( 3 . 2 1 w i l l be regarded as a d e f i n i t i o n .
It i s now necessary t o o b t a i n some u s e f u l thermodynamic r e l a t i o n s
Since t h e d e f i n i t i o n of entropy s a t i s f i e d by t h e thermodynamic v a r i a b l e J.
o f a p e r f e c t gas y i e l d s t h e equat ion:
t h e combination o f Equat ion ( 3 . 2 ) w i t h the above r e l a t i o n y i e l d s :
Hence, we o b t a i n the t h r e e r e l a t i o n s :
[%Is = - I ; 2 [g] = - c P ; B 2 [s) = - Fl3
P P J P
* See Concluding Remarks
-.l 2-
( 3 . 4 )
From t h e l a s t two equat ions i t fo l lows t h a t :
[%Ip = [s], Likewise, f rom Re la t i on ( 3 . 2 ) one gets
(3.5)
But, from the d e f i n i t i o n o f t he non- isen t rop ic propagat ion speed a
! 2 . 7 ! 1 g iven by
(Equat ion
i t f o l l o w s , us ing Equation (3.6) and r o u t i n e thermodynamic r e l a t i o n s , t h a t
7 a
( 3 . 7 )
f o r a J = cons tan t process.
Equat ion ( 3 . 7 ) i n d i c a t e s t h a t the parameter tends t o i n f i n i t y as
the process under cons ide ra t i on approaches an i s e n t r o p i c process i n the l i m i t .
Th is p r o p e r t y w i l l be confirmed i n subsequent d e r i v a t i o n s by a c t u a l l y de f i n ing
the parameter B i n terms o f boundary values.
Again us ing the d e f i n i t i o n o f t h e J f u n c t i o n i n Equation ( 3 . 2 ) t he
f o l l o w i n g r e l a t i o n ho lds
and, by us ing Equation (3.71, t h i s becomes
-1 3-
Hence, using the f i r s t of Equation ( 3 . 4 ) i t follows t h a t
o r , using Relation (3.7)
[%Is = - $3
In a s imilar manner one deduces from Equation (3.8) t h a t
2
[2jP = -
(3.9)
( 3 . 1 0 )
Consider the function G and i t s d i f fe ren t ia l as defined i n Equation ( 2 . 1 7 ) :
Combining the two i t follows t h a t
From the resul ts obtained i n Equation (3.4) this may be written
T h i s l a s t relation implies t h a t
Consider next the second of Equations (3.3)given by
(3.11)
(3.121
Using Equation ( 3 . 7 ) i t f o l l o w s t h a t
From t h i s r e s u l t we o b t a i n t h e r e l a t i o n s
( 3 . 1 4 )
( 3 . 1 5 )
It w i l l a l s o be use fu l t o cons ider the pressure as a f u n c t i o n o f
d e n s i t y and the J - v a r i a b l e and subsequently l e t t h e d e n s i t y be a f u n c t i o n o f
J and S [ t h i s procedure was used t o ob ta in the governing d i f f e r e n t i a l
Equat ion ( 2 . 1 9 ) 1. A r o u t i n e comparison o f c o e f f i c i e n t s by i nvok ing l i n e a r
i ndependence y i e l ds the re1 a t i ons :
o r
I n a s i m i l a r manner
( 3 . 1 6 b
-1 5-
or
( 3 . 7 7 6 )
and these may be e a s i l y v e r i f i e d by t h e use o f Table I .
I f we now consider t h e J - f u n c t i o n d e f i n i t i o n and the combined f i r s t
and second l a w equat ion o f thermodynamics, t h e f o l l o w i n g r e l a t i o n i s
obtained :
L - Equation ( 3 . 1 8 ) r e s u l t s i n t h e f o l l o w ng r e a t i o n s :
( 3 . 1 8 )
Some o f t h e above r e l a t i o n s which may be u s e f u l f o r subsequent d e r i v a t i o n s
a r e c o l l e c t e d i n Table I .
-1 6-
I
I- - Q
ll
I-
z 0 U a
“a I CJQ
L
u z 3 LL I
- pa P
-Iv) rmlm- J
w I I-
c, S l-a c, v) S 0 u
=a Pa I
II
Q
v, W Ln Ln w V 0 r x n
W I I-
a z
?
W
I- H 1
“&“ I
I1
’7
I- 1
ll
-l
c3v) x c -I w 1 m a I-
-
=I‘ a c, v) 5 0 u
c? I Q II
v)
h
I
ll
I
I1
v)
-1 7-
4. COMPARISON WITH EXISTING THEORIES
The d e r i v a t i o n o f t h e governing equat ion i n terms of two thermodynamic
var iab les S and J and the subsequent d e f i n i t i o n of t he J - f u n c t i o n as
presented i n Sect ions 2 and 3 respec t i ve l y , makes i t poss ib le t o compare the
der ived equat ion w i t h those o f e x i s t i n g theo r ies and show t h e i r equivalence.
L e t us cons ider Equation (2 .19) which was shown t o be
( 4 . I I
The func t ions ff and K were d e f i n e d i n Equation (2 .17) and were g iven by:
fl = ( I + [%)J) K = - [s)s ( 4 . 2 )
-1 8-
However, f rom the r e l a t i o n s i n v o l v i n g the J - func t i on i n Table 1 we i n f e r
t h a t
Moreover, f rom Equat ion (3.8) these may be w r i t t e n :
! i = 7 B 2 ;
a (4.4)
Using these values o f ff and K i n Equation (4.1), the f o l l o w i n g form i s
obtained:
n
V'S - a ~ S - ~ ( b n a ) m 2 2 U 2 vs = z
We can now d i v i d e t h i s Equation by a2 no t ing t h a t :
-1 9-
I u2s I u I us a a a
Under these condi ti ons, Equat ion (4.5) becomes
& [7m] I us - = -
F i n a l l y , the c o e f f i c i e n t o f the g r a d i e n t of the & f u n c t i o n may be
w r i t t e n :
( 4 . 7 )
-20-
Hence, Equat i on (4.7) becomes :
Equation (4.9) r e p r e x i i t s the f i n a l f e z nf Equation (2.19), us ing the
thermodynamic J f u n c t i o n as de f i ned i n Equation (3.2) . It may be looked
upon as a convected n o n l i n e a r wave equat ion i n a non - i sen t rop i c reg ion i n
which t h e f o r c i n g f u n c t i o n i s represented by t h e r i gh t -hand s ide o f t he
equa ti on.
A l t e r n a t e l y , i n a formal manner, one could l ook upon Equation (4.9) as
a convected wave equat ion i n a reg ion i n which the J f u n c t i o n i s propagated,
i t s f o r c i n g f u n c t i o n being given by en t ropy generat ion, viscous e f f e c t s ,
body fo rces and v e l o c i t y f l u c t u a t i o n .
Equation (4.9) i n the a l t e r n a t e form:
I n t h i s case i t i s convenient t o w r i t e
-21-
The l a s t form o f the equat ion i s useful when a non- isen t rop ic r e g i o n i s
imbedded w i t h i n a l a r g e r r e g i o n of qu iescent f l u i d .
case when a l a r g e ampl i tude pressure wave i s propagated i n t o a qu iescent
space by h i g h l y d i s t u r b e d non- isen t rop ic f l o w i n c l u d i n g moderate shock waves.
I n t h i s case the r igh t -hand s i d e o f Equation (4.10) represents the non-
i sen t r o p i c f o r c i n g f u n c t i on.
This i s obv ious ly t h e
I n t h e case when the J-wave propagated by Equation (4.10) i s generated
by i s e n t r o p i c processes, t h e ent ropy d e r i v a t i v e s vanish f rom the equat ion.
Also, the J - func t ion v a r i e s as t h e negat ive pressure f u n c t i o n s ince i t was
shown tha t * :
[g)s = - I (4..111
2 I t w i l l a l s o be shown t h a t , f o r cons tan t ent ropy, t h e propagat ion speed a
becomes the speed of sound a2 i n the 1 i m i t.
Under these assumptions, Equat ion (4.10) takes the f o l l o w i n g form:
( 4 . 1 2 )
*Refer t o Table 1.
-22-
forms used i n c l a s s i c a l i n v e s t i g a t i o n s o f the
It should be noted, however, t h a t t h i s equiva
i s e n t r o p i c pressure v a r i a t i o n s are pos tu la ted
c iass ica ; i n v e s t i g a t i s n s . The present analys
Equation (4.12) i s the d i f f e r e n t i a l equation de r i ved by P h i l l i p s
L i g h t h i l l ' s fo rm o f ( P h i l l i p s , 1960) w i t h t h e en t ropy terms deleted.
Equat ion may be ob ta ined by us ing i s e n t r o p i c r e l a t i o n s between t h e
pressure and d e n s i t y func t ions ( P h i l l i p s , 1960, a l s o Ribner, 1962).
This d e r i v a t i o n i n d i c a t e s the equivalence o f Equat ion (2.19) t o the
aerodynamic no ise problem.
ence hc lds t r u e o n l y when
as i t i s i n the case o f most
which i s aimed a t an
the i n c l u s i o n o f moderate
c sound generat ion, pu ts
I t assumes those the emphasis on the d i s s i p a t i v e terms o f the equat ion.
terms t o be t h e main c o n t r i b u t i o n t o the f o r c i n g f u n c t i o n o f the wave
equat ion causing the propagat ion o f a f i n i t e ampl i tude pressure wave.
S
i n v e s t i g a t i o n o f entropy-producing regions w i th
shock waves and t h e i r c o n t r i b u t i o n t o aerodynam
-23-
5. SOME ADDITIONAL ASPECTS OF THE GOVERNING EQUATION
The form o f Equat ion (4.101 obta ined i n the preceding s e c t i o n makes i t
apparent t h a t one of t h e f o c a l p o i n t s o f t h i s a n a l y s i s i s t h e choice o f t h e
form o f t h e thermodynamic J - func t ion , which a l lows one t o draw an analogy
between the general form o f t h e der ived equat ion w i t h those used i n c l a s s i c a l
s tud ies o f the aerodynamic no ise problem.
analogy e x i s t s o n l y when i s e n t r o p i c v a r i a t i o n s o f t h e J - f u n c t i o n are al lowed.
Moreover, i t has been shown t h i s
A more general approach t o t h e problem, i n view o f the form o f the
der ived equat ion, seems t o i n d i c a t e t h a t , i n a r e g i o n a d m i t t i n g an a r b i t r a r y
thermodynamic process hav ing t h e form
t h e propagat ion o f J and S waves are m u t u a l l y dependent so t h a t ent ropy
generat ion causes a J-wave emiss
but ions t o those phenomena be ing
v e l o c i t y f l u c t u a t i o n s . I n a d d i t
coupled t o the remaining e q u a t i o
on (and
made by
on, t h e
s o f mo
v i ce-versa) w i t h a d d i t i o n a l c o n t r i -
t h e viscous e f f e c t s , body fo rces and
obta ined Equat ion ( 4 . 1 0 ) i s s t i l l
consider the poss i b i 1 i t i e s o f uncoupl i n g
ion . It would thus 'be useful t o
these equat ions by i d e a l i z i ng the
process t o s t r a i g h t l i n e s i n the J-S plane, a procedure which would be
p a r t i c u l a r l y s u i t a b l e f o r t h e a p p l i c a t i o n o f t h i s a n a l y s i s t o t h e propagat ion
c h a r a c t e r i s t i c s o f a f i n i t e ampl i tude pressure wave.
consider t h e two reg ions, a h i g h l y per tu rbed r e g i o n w i t h a p o s s i b l e
appearance o f moderate shock waves which i s imbedded i n a r e g i o n o f qu iescent
f l u i d , t h i s second quiescent r e g i o n a d m i t t i n g of f i n i t e ampl i tude acous t ic
waves caused by t h e per tu rbed reg ion.
For i t i s f e a s i b l e t o
-24-
This idealization implies, basically, a f a r f i e ld noise analysis since
near f i e l d e f f ec t s must admit a mixed J-S process taking place as represented
by Equat ion (5.1).
I t i s apparent from the previous analysis and the obtained resu l t s t h a t
entropy changes would predominate in the perturbed region due t o the highly
diss ipat ive character of these processes (e.g., shock wave appearances). In
sixh a case the pressure and density functions would be primarily dependent
upon the entropy s t a t e s o f the f lu id particles and the thermodynamic process
may conceivably be approximated by J=constant and S varying. I n terms
of the previous analysis the d i f fe ren t ia l equation governing such a
diss ipat ive region would be given by
I n other words, in a region in which velocity fluctuations, viscous
e f fec ts and non-conservative body forces predominate, an S-wave i s generated
whose propagation character is t ics are largely determined by the non-isentropi c
speed a.
due t o instantaneous velocity o r viscous changes, entropy effects will a lso
be generated and must be considered as indicated by the form of Equat ion (5 .21 .
Moreover, in the presence of sudden discontinuities in the region
On the other hand, when the quiescent region i s considered, the
assumption of f a r f i e l d effects would also cal l fo r the additional s t ipulat ion
o f isentropy.
the govern ing Equation ( 4 . 1 0 ) reduces simply t o :
In t h i s case, excluding t h e perturbed region from consideration,
-25-
(5 .31
Here, t h e functions? value o f tends t o the i s e n t r o p i c propagat ion speed n
a', as entropy changes approach zero.*
F i n a l l y , when bo th reg ions a r e considered simultaneously, w i t h the
d i s s i p a t i v e reg ion p l a y i n g the r o l e o f a f o r c i n g func t i on propagat ing acous t i c
waves i n t o t h e quiescent reg ion , t h e governing equat ion ( 4 . 1 0 ) takes on the
f o l l o w i n g form:
I - ( 5 . 4 1
I t should be noted, f i n a l l y , t h a t i n {he more general case where no
simple assumption o f two d i s t i n c t thermodynamic reg ions can be made and
where t h e respec t i ve paths o f t h e thermodynamic processes i n t h e J-S plane
do not f o l l o w s t r a i g h t l i n e s , as i s c nce ivab le i n these extreme phys i ca l
cond i t ions , t h e analogy between Equat on ( 4 . 1 0 ) and t h a t de r i ved above cannot
be drawn a phi&.
*See Sect ion V I .
-26-
6. THE PROCESS J = CONSTANT
I n accordance w i t h the preceding der iva t ion , t h e r e g i o n i n which
d i s s i p a t i v e mot ion predominates w i l l be approximated by t h e thermodynamic
process J=constant. T h i s r e g i o n , i n which entropy v a r i a t i ons predominate,
permi ts thermodynamic v a r i a b l e s which are, by hypothesis, f u n c t i o n s o f t h e
space coord inates and t i m e through the medium o f entropy.
Using Equat ion (3.61 i t i s i n f e r r e d t h a t
[g] - $ = 7 I (6.7 1
I n t e g r a t i o n o f t h i s equat ion y i e l d s t h e pressure d e n s i t y r e l a t i o n f o r t h e
J=constant process:
I V I t
p = y = l - - p + A P
ng t h e i n t e g r a t i o n constant (Figure 1 ).
i n g Equat ion (6 .2 ) w i t h the entropy equat ion f o r a p e r f e c t gas
w i t h A be
coup
(6.21
I S-So P = p-B Y U P ( - T) (6 .31
where B i s a constant , y i e l d s t h e pressure as a f u n c t i o n o f entropy f o r
-27-
PRESSURE RATIO
4.21 I I
PRESSURE RATIO
6
PRESSURE RATIO
Figure 1 . Pressure vs Pensity for JnConstant Process kpplied to Shock Laver Transitton Flow
-28-
L e t pi and pi r e f e r t o the i s e n t r o p i c values of these func t i ons
when S = S . (i = 0 , I ) respec t i ve l y . Under these cond i t i ons the cons tan t
8 i s determined i n terms o f e i t h e r i n i t i a l or end c o n d i t i o n s (Equation ( 6 . 3 ) ) .
L
From t h e c o n d i t i o n t h a t
P = Po when s = so (6.5)
t h e cons tan t r a t i o $ i n Equation (6.4) i s a lso determined. The r e s u l t i n g
expression f o r t he pressure f u n c t i o n takes on t h e f o l l o w i n g form:
Y
(6.61
2 Here a. i s t he i s e n t r o p i c propagation speed when S=So. The constant
parameter 6' i s now evaluated from the cond i t i on t h a t p=pl when S = S l
a t the end o f t h e process
2 B =
-j - '1 (6.71
It may be shown i n a s p e c i f i c app l i ca t i on* t h a t , when no en t ropy pro-
*For moderate shock layers , t h e numerator and denom
a re expressed i n terms o f t he upstream Mach number
i t i s found t h a t , as E + 0, E +
0
n a t o r o f Equation (6 .7 )
M;. S e t t i n g M~ = I + €
.e., as S, -+ So,
2
B~ + - m (see Sect ion 7 ) .
-29-
duct ion occurs, i .e . , when S I -+ So i n the l i m i t , t he va lue of t h e parameter
8' tends t o minus i n f i n i t y i n t h e l i m i t . Th is p roper t y makes Equat ion ( 6 . 1 )
i s e n t r o p i c when S I -+ So, as expected.
On t h e o the r hand, once a non- isen t rop ic , J=constant process takes
place, i .e. , S I > So, the parameter 8' becomes f i n i t e . Under these con-
d i t i ons i t s ex is tence and constancy serve t o under l i ne both en t ropy p roduc t i on
and i r r e v e r s i b i l i t y o f the process.
f i n i t e i t r e f l e c t s the process o f en t ropy p roduc t i on and i t s constancy
determi nes the i r r e v e r s i b l e cha rac te r o f t h e event.
For once the parameter 8' becomes
To eva lua te the d e n s i t y p as a f u n c t i o n o f en t ropy f o r t he process
J = constant, Equations (6 .3 ) and (6 .6 ) a re used t o y i e l d :
1
s-So P = P o (6.81
Equation ( 2 . 7 ) combined w i t h Equations (6 .6 )andh(6 .b ) y i e l d s the
temperature T as a f u n c t i o n o f en t ropy f o r t h e process J=constant.
It i s apparent from Equat ion 16.91 t h a t , as t h e en t ropy s tends t o
T tends t o T which i s t he i s e n t r o p i c value o f t h e temperature so , 0'
func t ion . * I n a s i m i l a r manner, when S tends t o S I , T tends t o T I .
These can be r e a d i l y v e r i f i e d by us ing Equations (6.3), (6.71 and (6.91
*Note tha t , as S-So -+ E ,
(See Section 7) .
=
-30-
The preceding d e r i v a t i o n s a l l ow t h e determinat ion o f t h e non-
i s e n t r o p i c propagat ion speed a2 as a funct ion of entropy. Using
Equations ( 3 . b ) , (6 .6 ) and (6.8) i t i s apparent t h a t :
(6.70)
Tije n ~ n - i s e n t r o p i c propagat ion speed a2 i s p l o t t e d f o r d i f f e r e n t entropy
p roduc t i on values i n F igure 2.
In t h e l i m i t i n g case when S tends t o S o , a2 tends t o the
i s e n t r o p i c propagat ion speed a i , as ind i ca ted by the form o f Equation
[ 6 . 1 0 ) . * Here i t i s o f i n t e r e s t , however, t o consider the value o f t he
propagat ion speed a = a1 when S = S I , i.e., a t t h e t e r m i n a t i o n o f t he
non- i sen t rop i c JI=constant process. To t h i s end, Equat ion (6.10) i s w r i t t e n
r i n the form
n 0
1 . - 2
Equations (6 .3 ) and ( 6 . 7 ) are then used t o o b t a i n t h e f o l l o w i n g
s u b s t i t u t i o n s :
1 -SO
I - - 2 (2) = [.XP (- T)] ( 6 . 1 2 ~ )
2 *When n o J =constant process occured, i .e . , when I B I -+
-31 -
1.6-
1.4
M, = 1.5
-
ENTROPY 2 C P
2.5
2.0
F igure 2. The non- isent ron ic Propaaat ion Speed a v s Entropy t o Shock Laver T r a n s i t i o n Flow
M, = 2.0
-
-32-
0.
Appl i ed
and
Under these c o n d i t i o n s Equat ion
2 2 " 1
= 2 a,
( 6 . 1 2 b J
6 , 1 1 1 s i m p l i f i e s t o
( 6 . 131
I n t h i s case, however, t h e parameter 8' i s f i n i t e , s ince a n o w
i s e n t r o p i c process took place.
process does n o t reduce t o i t s corresponding i s e n t r o p i c speed, b u t i s
Thus t h e propagat ion speed a t t h e end o f t h e
m o d i f i e d by a f a c t o r 1
J7$ It w i l l subsequently be shown t h a t , f o r small ent ropy changes, t h i s f a c t o r
i s c l o s e t o u n i t y . *
It appears f rom the above considerat ions t h a t when a process
J=constant occurs i n which ent ropy changes predominate, the thermodynamic
v a r i a b l e s (p, p , T, C( , etc . ) are funct ions o f t h e h i s t o r y o f t h e process
r a t h e r than t h e instantaneous values o f the s t a t e o f the f l u i d p a r t i c l e s
2
(no te t h e reference t o S0 as t h e i n i t i a l s t a t e t o which a l l values o f the
*For l a r g e ent ropy v a r i a t i o n s ( M > 5.01 t h e above statement does n o t hold.
-33-
variables are referred) . Moreover, whereas a l l functional values of the
pressure, density and temperature are continuous a t the boundary of the
region in which the process takes place, t h i s i s no t so fo r the derivatives
of these functions.
For i t i s noted t h a t a l l functions take on t h e i r isentropic values
on the boundary of the region, excepting the propagation speed a, which
differs from i t s isentropic counterpart a t the termination of the process
by a fac tor which i s d i rec t ly dependent upon the amount of entropy production
tends t o unity as the entropy production tends t o
l y appears t h a t a non-isentropic region may have a
due t o a difference in the propagation speeds when
c region. However, the functional values of the
thermodynamic variables are no t discontinuous.
during the process and
zero.
I t then forma
di sconti nuous boundary
i t adjoins an isentrop
7. THE CASE OF MODERATE SHOCK WAVES
The preceding analysis lends i t s e l f t o simplif5cation when the
entropy production region under consideration i s the region of a moderate
shock wave.
To evaluate the functional variations of the thermodynamic variables
in t h e shock layer as a function of entropy production, i t i s convenient t o
evaluate i ni t i a1 ly cer ta i n recurring expressions appearing i n the preceding
analysis and given by
-34-
Here s t a t e zero.denotes the value o f t h e v a r i a b l e s i n f r o n t of the
shock l a y e r , and s t a t e one r e f e r s t o t h e i r va lue behind t h e shock l a y e r .
From the shock t r a n s i t i o n equations i t i s ev ident t h a t
(7.31
( 7 . 4 1
2 L e t us consider the value o f the parameter 6 . It i s apparent
f rom R e l a t i o n (6.7! t h a t
Y - 1
(7.51
I n the l i m i t , when S I tends t o So, t he normal component o f t he
Mach number upstream o f t he shock l a y e r tends t o u n i t y .
Equat ion (7.5) may be evaluated i n the l i m i t by s e t t i n g
Consequently,
Mobin 2 2 e = 1 + E (7.61
-35-
Fol lowing t h i s procedure i t can r e a d i l y be shown t h a t :
and
(7.7cl
Hence, as SI +. S o , i .e. , as E approaches zero ( t o t h e f i r s t approximat ion)
T h i s i s t h e c o n d i t on which makes t h e J=constant process i s e n t r o p i c i n t h e
l i m i t , i n accordance w i t h prev ious d e r i v a t i o n s [see Re la t ions (3 .81 and
(6 .71 1. The parameter 6' i s p l o t t e d as a f u n c t i o n o f Mach number f o r a
g iven shock t r a n s i t i o n process i n F igure 3.
I n a s i m i l a r manner, u s i n g t h e shock t r a n s i t i o n r e l a t i o n s , t h e
values o f t h e f u n c t i o n s p and p versus ent ropy a r e p l o t t e d i n F igures 4
and 5.
It should a l s o be noted t h a t most o f t h e d e r i v e d r e l a t i o n s f o r
pressure, dens i ty , e tc . , c o n t a i n the f a c t o r
-36-
E = B 2 { exp (- s>- 7 -% I ) ( 7 . 9 )
negat i ve i n f i n i t y as (- f ) [see Equat 3 i n Equat ion ( 7 . 9 ) tends t o zero as ( - E
t o t a l express ion ir! Equat ion ( 7 . 9 ) tends
S -+ S I ) as expected.
Now i n t h e l i m i t as SI tends t o So, the parameter 8' tends t o
on (7.811, whereas t h e second f a c t o r
[see Equat ion (7.711. Hence, t h e
t o zero when SI -f So (and
The f o l l o w i n g impor tan t consequence of the above d e r i v a t i o n should
be noted.
process , the non- isen t rop i c e f f e c t s cannot be a r b i t r a r i l y disregarded,
f o r i t c o u l d be reasoned t h a t , even f o r the case o f weak shocks, t h e ent ropy
v a r i a t i o n i s o f a t h i r d - o r d e r magnitude, and t h e r e f o r e i t s e f f e c t s c o u l d
conceivably be disregarded.
cons idera t ions p o i n t t o t h e f a c t t h a t t h i s reasoning may n o t be j u s t i f i e d .
This i s due t o t h e f a c t t h a t , even though the ent ropy v a r i a t i o n i s o f a
t h i rd -order magnitude, i t s product w i t h the process parameter
as a f i r s t - o r d e r magnitude [Equation (7.911.
o f pressure, densi ty , temperature, etc., i n t h e non- isen t rop ic r e g i o n a l s o
vary as a f i r s t - o r d e r magnitude. Moreover, t h e d i f f e r e n t i a l equat ion
governing t h e aerodynamic sound generation, as der ived i n Equat ion (5 .41 has
a l s o f i r s t - o r d e r v a r i a t i o n o f the non- isentropic terms.
c o n t r i b u t i o n o f the non- i sen t rop i c terms cannot be d isregarded w i t h i n the
framework o f t h i s a n a l y s i s and t h e entropy produc t ion terms must be taken i n t o
account even f o r smal l v a r i a t i o n s of the ent ropy f u n c t i o n s i n t h e process
considered (e.g., weak shocks).
Even when a smal l ent ropy change occurs dur ing any a r b i t r a r y
On the o t h e r hand, t h e present t h e o r e t i c a l
8' v a r i e s
Thus, a l l the func t iona l values
Consequently, t h e
-37-
0
-50
-100
-150 1 2 3
MACH NUMBER
4
2 2 Figure 3. The Darameter 6 /a s Upstream Fach Number f o r J=Cons tan t Process Anolied t o gi Normal Shock T r a n s i t i o n Flow.
- 38-
1 . 2 0 .
1.16 - 0 n.
k
w
n.
0 4 8 12 16 20 24 26
ENTROPY (s - sO)ic, ix &j
4 . 6 1 1
2.0
1 .8 -
0
Y
ENTROPY (s - so)/,-- (x 10-3) -r
ENTROPY (S - So)/cp (X ENTROPY (S - S )/ (X O CP
F i q u r e 4 . Shock L a y e r T r a n s i t i o n Flow.
Pressure vs . Entronv f o r J=Constant Process ADnlied t o
-39-
0 8 16 24 ENTROPY (s - sO)/cp (x 10-5)
2.8
2.6 - 2.4 -
MACH NO. = 2.0
0 4 8 ENTROPY (5 - So)/cp (X 10-2)
n MACH NO. = 1.4
0 4 8 12 ENTROPY (s - sO)/cp (x 10-3)
4.6'
4 .2 -
MACH NO. = 3.0
0 8 16 24
ENTROPY (S - SO)/cp (X IO-')
Fiqure 5. Densitv v s . Entropv f o r J=Constant Process AnDlied t o Shock L a y e r Transition Flow.
-40-
11. ENGINEERING APPLICATION
8. INTRODUCTION
A s i m p l i f i e d eng ineer ing model o f t h e p ropagat ion c h a r a c t e r i s t i c s of
a f i n i t e ampl i tude pressure wave i n high-speed f l o w w i t h moderate shock wave
fo rmat ion i s presented i n the subsequent sections. The ac tua l computations
o f the r e s u l t s were based i n i t i a l l y on a previous model (Peter and L i , 1965
i n which o n l y non- isen t rop ic changes o f the pressure f u n c t i o n were coii-
s idered, s ince a t t h a t t ime some o f t he r e s u l t s o f t h e present ana lys i s cou
n o t be i nco rpo ra ted i n t o the computer programs.* Nevertheless, the bas i c
c h a r a c t e r i s t i c s o f the d e r i v a t i o n s are s i m i l a r i n bo th cases and serve t o
i l l u s t r a t e some phys i ca l aspects o f non- isen t rop ic f o r c i n g func t i ons i n
aerodynamic no i se generat ion.
I n accordance w i t h the t h e o r e t i c a l aspects o f the preceding ana lys i s
an entropy-produci ng reg ion imbedded i n an i sen t rop i c medi um w i 11 produce
acous t i c e x c i t a t i o n s in the f a r f i e l d . I n the near f i e l d a mixed J-S process
takes place** w i t h a r e s u l t i n g en t ropy wave emission.
process w i l l n o t be considered here.
e f fec t o f f i n i t e ampl i tude wave propagation i n t o a quiescent i s e n t r o p i c
reg ion .
The near f i e l d mixed
Emphasis i s p laced main ly on the
* The o n l y d i f fe rence cons is t s i n t h e Mach number dependence of t he wave
ampl i tude. Th is adds a constant fac to r t o the dec ibe l count of t he
r e s u l t i n g i n t e n s i t y .
** See Sect ion 5.
d
-41 -
The governing equations f o r t h e propagat ion phenomena have been
der ived i n Equations ( 5 . 2 ) and (5.41. Equation ( 5 . 2 ) i n d i c a t e s t h a t non-
conservat ive e f f e c t s o f f l u i d motion, i .e., r o t a t i o n a l v e l o c i t y f l u c t u a t i o n s ,
viscous e f f e c t s and d i s s i p a t i v e body fo rces cause an en t ropy wave propagat ion
i n the per tu rbed reg ion .
region i s imbedded i n an i s e n t r o p i c qu iescent medium, an acous t i c e x c i t a t i o n
Equation (5.4) i n d i c a t e s t h a t , when the per tu rbed
o f the qu iescent reg ion a l s o takes place.
I n t h e present eng ineer ing approach the th ickness o f t he t r a n s i t i o n
l a y e r i s d isregarded and a discont inuous jump i n the value o f t he f o r c i n g
f u n c t i o n i s assumed.
through a s e r i e s o f d isturbances caused by random d i s t r i b u t i o n o f impulses
imparted t o the shock l a y e r .
t h a t entropy c o n t r i b u t i o n s t o the f o r c i n g f u n c t i o n predominate w h i l e ve loc
f l u c t u a t i o n s a re o f second o rde r magnitude even f o r t he case o f weak shock
waves. T h e i r e f f e c t on the f o r c i n g f u n c t i o n seems t o be i n d i r e c t by impar
i n g i n i t i a l e x c i t a t i o n s t o the en t ropy f u n c t i o n and thus t r i g g e r i n g the
mechanism of emission.
9.
The t ime dependence o f t he f o r c i n g f u n c t i o n en te rs
An order o f magnitude ana lys i s i n d i c a t e s
EN G I NE E R I N G REP RES ENTAT I ON
I n accordance w i t h t h e preceding discussion, t he governing d i f f e r e n t i a l
equation w i l l now be a p p l i e d t o analyze the propagat ion c h a r a c t e r i s t i c s o f
moderate shock l a y e r s i n supersonic nozz le f low.
I t was shown i n Sec t ion 5 t h a t t he equat ion rep resen t ing non-
i s e n t r o p i c wave propagat ion i n t o a qu iescent reg ion i s given by:
-42-
The case o f 5 shock l a y e r i s now considered. To do t h i s i t i s
necessary t o s p e c i f y t h e r igh t -hand s i d e o f Equat ion (9.71, p a r t i c u l a r l y the
ent ropy f u n c t i o n and i t s gradients . A r igorous f u n c t i o n a l e v a l u a t i o n o f
the remaining equat ions o f motions
which Equat ion ( 9 . 1 ) i s s t i l l coup
w i l l be necessary t o represent the
these v a r i a b l e s cannot be accomplished, however, w i t h o u t s o l v i n g s imul taneously
[see Equations ( 2 . 7 ) t o (2 .411 w i t h
ed. To c i rcumvent t h i s d i f f i c u l t y i t
J-constant process as i f i t i s t a k i n g
p lace i n a r e g i o n o f zero th ickness, so tha t a sudden d iscont inuous jump i n
t h e dependent v a r i a b l e s across the shock l a y e r occurs. This representa t ion
ascr ibes t o t h e remaining equat ions o f motion a c o n t r i b u t i o n whose mean
square va lue i s g iven by t h e Rankine-Hugoniot r e l a t i o n s .
formal d e r i v a t i o n of t h e governing equations has i m p l i e d t h e ex is tence and
However, t h e
c o n t i n u i t y of t h e dependent v a r i a b l e s up t o and i n c l u d i n g second d e r i v a t i v e s ,
Th is r e s t r i c t i o n l i m i t s the a n a l y t i c a l representa t ion o f the d iscont inuous
charac ter of t h e dependent var iab les t o those f u n c t i o n s whose d e r i v a t i v e s
admit a n a l y t i c a l c h a r a c t e r i s t i c s . For t h i s reason genera l i zed f u n c t i o n s
-43-
have been employed t o represent t h e f o r m a l i s t i c behavior o f t h e jump con-
d i t ions dur ing t r a n s i t i o n .
whose successive d e r i v a t i v e s admit a n a l y t i c a l i n t e r p r e t a t i o n s .
T h e i r g rad ien ts become D i rac D e l t a funct ion:
I t w i l l be shown subsequent ly t h a t t h i s d iscont inuous charac ter o f
t h e dependent v a r i a b l e s d u r i n g t h e process of t r a n s i t i o n r e s u l t s i n a r a d i a t i n g
acoust ic d i p o l e f i e l d . * The r a d i a t i n g d i p o l e s are d i s t r i b u t e d over t h e
shock l a y e r and t h e i r l o c a l s t r e n g t h i s p r o p o r t i o n a l i n t h e mean t o the
Rankine-Hugoniot t r a n s i t i o n values.
A no tab le s i m p l i f i c a t i o n o f t h e mathematical d e r i v a t i o n s may be
achieved by imposing upon t h e r a d i a t i n g f i e l d t h e cond i t ion** t h a t t h e t ime
dependence o f t h e f o r c i n g f u n c t i o n w i l l e n t e r through a s e r i e s o f d is turbances
caused by a random d i s t r i b u t i o n o f impulses imparted t o t h e shock layer .
The time d e r i v a t i v e s o f these impulse f u n c t i o n s a r e m u l t i p l i e d . b y mean
t r a n s i t i o n values which determine t h e ampl i tude o f the propagated wave and
which may be regarded as s l o w l y v a r y i n g cont inuous t ime func t ions w i t h
n e g l i g i b l e gradients .
funct ions may be neglected i n the f i r s t approximat ion i n view of t h e f a c t
t h a t no t ime i n t e g r a t i o n appears i n t h e s o l u t i o n t o t h e wave equat ion.
P h y s i c a l l y , t h e assumption i m p l i e s t h a t mean t r a n s i t i m values of the
f o r c i n g f u n c t i o n determine t h e ampl i tude o f t h e e m i t t e d waves, whereas t h e
random t ime impulses supply t h e mechanism t r i g g e r i n g t h e emission.
As a consequence, t h e t ime d e r i v a t i v e s o f t h e impulse
* Sect ion 10.
** This c o n d i t i o n i s s i m i l a r t o t h a t imposed on the f o r c i n g f u n c t i o n used
i n computations (Peter and L i , 1965, a l s o Sec t ion 8).
-44-
Under these cond ens, Equation ( 9 . 1 1 becomes
( 9 . 2 1
An order-of-magni tude ana lys i s o f t he f o r c i n g f u n c t i o n of Equation
(9.21 w i l l now be conducted. I n the first place, s ince t h e reg ion o f t he
shock i n which the J=constant process takes place i s represented by a l a y e r
o f v a n i s h i n g l y smal l th ickness, t he viscous term i n t h e equat ion may be
accounted f o r by the jump across the layer t o which the a c t i o n o f the
viscous fo rces i s conf ined.
such a d iscont inuous rep resen ta t i on o f t h e f o r c i n g f u n c t i o n generate a
quadrupole r a d i a t i o n f i e l d which i s o f t h e nex t o rde r o f magnitude). I n
a d d i t i o n , t he body forces do n o t appear under normal f l o w cond i t i ons and
( I t can be shown t h a t t h e v iscous terms i n
may be disregarded.
For the sake o f conven
t h e f u n c t i o n by Ui(i = 1 , 2 , 3
ence we w i l l designate the remaining terms of
(9.31
-45-
I t i s evident t h a t the two functions, w1 and U 2 represent the
W3 contribution of entropy production t o the forcing function, whereas
designates the interaction of velocity gradients ( f luctuat ions) and
represents t he i r contribution t o sound generation.
fluctuations will now be shown t o const i tute a second-order magnitude when
compared with the entropy terms o f the forcing functions, even fo r the case
of weak shock layers.
These mechanical velocity
To t ha t end, consider the case of one-dimensional flow and transform
the term W g by the use of Equations ( 2 . 2 ) , (2.73) and ( 4 . 4 ) t o o b t a i n
where Cui = U(x
Equation 9 .4 ) reduces t o
when mean intensi ty values are considered.
I n the entropy-producing region, i . e . , in
by the process J=constant, the propagation speed
isentropic sound speed ad fo r very weak shock
( 9 . 4 )
the shock layer represented
cx i s of the order of the
ayers . * Li kewi se , the
* Section 7.
-46-
v e l o c i t y U a l s o approaches son
magnitude. i
8 1 . , _
’ R e f e r r i n g now t o p rev ious
c speed* and i s o f t he same o rde r of
der iva t ions , i t was shown** t h a t
n
2 w i t h E = (MO - 7 ) and such** t h a t E < < 1 . It then f o l l o w s t h a t
(9 .71
Using these est imates i n the term
Equation (9.31, i t i s i n f e r r e d t h a t Wi o f the f o r c i n g func t i on , as i n
( 9 4 2 2 W3” 4UOE 1
Thus, t he present theory i nd i ca tes t h a t v e l o c i t y f l u c t u a t i o n s
(mechanical e f f e c t s ) i n a non- isen t rop ic reg ion are o f a second-order
magnitude when compared t o the en t ropy terms even f o r t he case o f weak shocks
when en t ropy v a r i a t i o n s , p e t b e , are very smal l .
* From above
** Sec t ion 7.
-41-
I t should be noted, however, t h a t such v e l o c i t y f l u c t u a t i o n s may
a f f e c t the non- isen t rop ic sound generat ion i n d i r e c t l y by i m p a r t i n g i n i t i a l
e x c i t a t i o n s t o the ent ropy f u n c t i o n , thus p r o v i d i n g t h e t r i g g e r i n g mechanism
necessary f o r emission.*
o f a f i x e d supersonic nozz le.
reg ion f rom which acous t ic exc
quiescent medium. The analogy
o f the emission mechanism. On
The shock l a y e r i s
10. AN ACOUSTIC ANALOGY
An acous t ic analogy i s now cons t ruc ted i n o rder t o eva lua te t h e pro-
pagat ion c h a r a c t e r i s t i c s o f a s t a t i o n a r y shock l a y e r fo rming i n t h e exhaust
regarded as a s t a t i o n a r y
t a t i o n s propagate n t o t h e surrounding
d isregards convect ve and d i s p e r s i v e e f f e c t s
t h e o t h e r hand, t h e l o c a l en t ropy grad ien ts
which predominate i n t h i s analogy were shown** t o be o f a comparable o rder
o f magnitude and represent a s u b s t a n t i a l p a r t o f t h e o v e r a l l problem.
The d i f f e r e n t i a l equat ion governing t h i s erni ss ion process may be
p u t , i n view o f Equat ion ( 9 . 2 ) and t h e subsequent d iscuss ions, i n t h e
f o l l o w i n g form:.
* See Equat ion ( 5 . 2 )
** See Equat ion (9 .8 )
-4a-
Here t h e en t ropy func t ion S has been w r i t t e n i n the form discussed
p r e v i o u s l y
( 1 0 . 2 1
I t i s noted t h a t t h e s p a t i a l dependence o f t h e en t ropy f u n c t i o n i s
represented by a product o f an ampl i tude func t ion A ( x 1 and the Heavyside
s tep funct ion. The ampl i tude f w c t i o n is propor t iona l i n the mean t o t h e
t r a n s i t i o n values o f entropy. I t s dependence upon s p a t i a l coord inates i s
in t roduced here t o take i n t o account thermodynamic non-equi 1 i b r i um f lows i n
t h e r e g i o n (e.g., curved shock l a y e r s ) . * For constant curvature shocks t h e
amp1 i tude f u n c t i o n depends upon t h e constant normal component o f the upstream
Mach number. The present i n v e s t i g a t i o n w i l l concentrate l a r g e l y on t h e
1 a t t e r case.
The shock sur face i s represented by t h e equat ion
= h(x ,y l (10.31
The s p a t i a l dependence i n Equat ion (10.31 w i l l imp ly a x i a l symmetry.
I n a c t u a l i t y t h e shock l a y e r s t r u c t u r e i n underexpanded supersonic
nozzles admits, i n c e r t a i n cases, o f a s l i g h t curva ture before the onset o f
t h e Mach d i s c (Peter and Kamo, 1963). However, i n t h e present fo rmula t ion ,
the curved e f fec ts w i l l i n t roduce unwarranted compl icat ions due t o t h e non-
e q u i l i b r i u m charac ter o f the f l o w behind the shock l a y e r a t the end o f the
non- isen t rop ic process ( i .e . , s p a t i a l entropy grad ien ts ) . For i n t h i s case
t h e process parameter 8' w i l l n o t r e t a i n i t s constant character s t i p u l a t e d
* I t can be shown t h a t such non-equi l ibr ium f lows make t h e thermodynamic
parameter 8' space-dependent.
-49-
here unless i t s average va lue i s used.
poss ib le Mach d i s c superpos i t ion should a f f o r d a good approximat ion, which
becomes exac t i n many cases.
Moreover, c o n i c a l s t r u c t u r e s w i t h a
Under these cond i t ions , the t o t a l ampl i tude o f t h e emission i s seen 2 t o depend upon t h e product ( 6 A ) independent of s p a t i a l v a r i a t i o n , w i t h
the upstream normal Mach number appear ing as a f l o w parameter.
The t ime v a r i a t i o n o f t h e ent ropy f u n c t i o n d ( , t ) , represent ing an
impulse-disturbance imparted t o the J=constant reg ion, w i l l be kepk i n a
general form
d(.t1 = - 2Ti ' 1 r ( w ) e-iddW -m
where r ( 0 ) i s t h e F o u r i e r t rans form of d ( X ) .
The s o l u t i o n o f Equat ion ( 1 0 . 7 ) may now be w r i t t e n
(10.41
W i-4 "0
e 02[i4 4
The i n t e g r a t i o n i s taken over t h e r e g i o n o f
2-h) ]dv ( 1 0 . 5 )
.* -
the d is turbance, *
* Here t h e rad ius 4
5 = observat ion p o i n t coordinates;
i s de f ined i n t h e usual manner h = I;.- ; I ; -+ -+ x = per tu rbed r e g i o n coord inates.
- 50-
Let us consider f i r s t the simplest case o f a normal shock layer
located a t some p o i n t z = ho, where h0 = constant. Under these conditions
the term v2H becomes a derivative of the delta function 6 ( z - h O ] with
respect t o Z.
the properties of the delta function):
As a consequence, Equation (10 .51 may be written(using
(70.6)
The integral i s taken over the surface u over which the disturbance takes
p l ace.
I t i s evident that the obtained solution represents a radiating dipole
f ie ld located a t
shock layer emitting pressure waves whose amplitude i s proportional t o the
t ransi t ion values of the Product ( A B ' ) .
f i r s t approximation the dependence of the acoustic intensi ty of radiation
upon the upstream Mach number.
z = hO. The radiating dipoles are distributed over the
The product determines t o the
From previous considerations i t i s readily
deduced t h a t :
-51-
2 aO
Y - 7
I
1 2 2 2YModin 8 - ( Y - 7 )
Y t l
7 7 -y
- 7
Here, MObine denotes the normal component o f t h e Mach number t o a shock
l a y e r i n c l i n e d a t an angle e=constant.
Equat ion (70.5) w i l l now be genera l i zed t o i nc lude shock l a y e r s o f
an a r b i t r a r y shape.
equation
Thus we l e t t he shock l a y e r surface be g iven by the
1 where $I denotes a fam i l y of g iven surfaces and $O=constant i s t h a t
p a r t i c u l a r sur face which defines t h e shock l a y e r .
Th is equation may a l s o be w r i t t e n z - b(x,y) = 0, on t h e shock; z - 6 =con-
s t a n t , otherwise.
-52-
For convenience we d e f i n e two mutual ly orthogonal f a m i l i e s o f
sur faces by
( 1 0 . 8 6 ) 2 2 3 3 F ( x , y , z ) = $ F ( x , y , z ) = J,
1 1 which are orthogonal t o F = J,
Equation ( 1 0 . 5 ) i s now transformed t o the c u r v i l i n e a r coord inates
J;" [i=1,2,31.
per tu rbed r e g i o n de f i ned by t h e vec to r x and n o t t o t h e p o i n t o f observat ion
which i s denoted by the vec to r 1. form:
It should he noted t h a t these coordinates r e f e r on l y t o the -+
Equation ( 1 0 . 5 ) i s now w r i t t e n i n t h e
m
where
i k h
G(z,;i = [ e ] i s a f u n c t i o n of and the $"s.
l10.10)
(10.11)
-53-
,
represents the Jacobian o f t rans fo rma t ion .
t he me t r i c c o e f f i c i e n t s o f t he c u r v i l i n e a r system.
The hi's (i = 7,2,3) represent
I n terms o f the coord inates $' i t i s ev iden t t h a t
( 1 0 . 1 2 )
7 s ince H = H(,$ ) on ly . Thus, Equat ion (70.9) becomes
[ 5'1 ' Z h 3 d+22dQ3 ( 1 0 . 1 5 )
h1
I f we denote the l i n e a r increments a long the c u r v i l i n e a r coord inate
l i n e s by ai(i = 1 , 2 , 3 ) so t h a t
Equat ion (70.15) can convenient ly be w r i t t e n i n the form
P"
( 1 0 . 1 7
- 55-
This l a s t equat ion c l e a r l y de f i nes a d i p o l e r a d i a t i o n f i e l d generated
I J I ~ . by a shock l a y e r whose form i s represented by the sur face
t h a t the d i p o l e a x i s p o i n t i n g i n the d i r e c t i o n o f
shock sur face.
i n t e n s i t y o f r a d i a t i o n o f a supersonic nozz le i n the presence o f shock
waves w i l l tend, i n ' the l i g h t o f t he present theory, t o be normal t o t h e
shock surfaces. Thus a con ica l shock whose angle w i t h respect t o t h e j e t
a x i s i s 45" w i l l r a d i a t e sound whose i n t e n s i t y w i l l t end t o be maximum along
t h e 45" a x i s o f t he j e t .
nozzles l i e i n the range o f 30" t o 60" i n c l i n a t i o n , i t appears t h a t t h e
maximum i n t e n s i t y o f r a d i a t i o n w i l l t end t o f o l l o w t h i s pa t te rn . *
It i s noted
l l i s normal t o t h e
I t i s then of immediate consequence t h a t the maximum
Since most shock l a y e r s i n underexpanded supersonic
I t should be noted t h a t t he r e s u l t a n t f i e l d o f r a d i a t i o n i s symmetric
w i t h respect t o the p ane passing through the o r i g i n and which i s normal t o
t h e axis o f t h e j e t .
t he condi t ions p reva i i n g upstream of t he shock wave were disregarded i n t h i s
analogy.
upstream cond i t i ons t o an e x t e n t which cannot be envisaged from the present
development.
be obtained from the ef fects of t h e terms disregarded here bu t -appear ing i n
the governing d i f f e r e n t i a l Equat ion ( 9 . 1 1 .
This i s due t o t h e f a c t t h a t the supersonic s t a t e o f
As a ma t te r of f a c t , backward r a d i a t i o n w i l l be impeded by the
A more accurate p i c t u r e of these e f f e c t s could, i t i s hoped,
*See Section 11 f o r a more complete d i scuss ion of con ica l shock propagat ion
c h a r a c t e r i s t i c s .
-56-
I t should a l s o be noted t h a t add i t i ona l sources o f r a d i a t i o n appear
i n ac tua l supersonic nozz le f l o w which could be, b u t a r e not, t r e a t e d by
t h e present approach.
boundary o f the nozz le f l o w a r i n g o f r a d i a t i n g d ipo les appears whose a x i s
i s d i r e c t e d a long t h e shock.
appear due t o the t a n g e n t i a l entropy d i f f e rence between the t e r m i n a t i n g shock
and the surrounding medium. These e f f e c t s , which must be t r e a t e d separate ly ,
w i l l n o t change the present d i r e c t i o n a l i t y and propagat ion c h a r a c t e r i s t i c s t o
a l a r g e e x t e n t due t o the orders o f magnitude o f t he respec t i ve sur face areas.
For instance, a t the t e r m i n a t i o n o f t he shock a t t he
Consequently, a d d i t i o n a l r a d i a t i n g sources
The appearance o f Mach d i scs i n the f l o w could be t r e a t e d by super-
p o s i t i o n b u t i t i s doub t fu l i f more use fu l i n f o r m a t i o n could be obta ined
w i t h o u t a r i go rous ana lys i s o f t he complete non- isent rop ic phenomena o f
h i g h speed f lows.
analogy.
p lace l o c a l l y .
due t o t h e i r geometr ical s i m i l a r i t y o f shape, which appears t o be o f a p re -
.dominant importance i n the l i g h t of t h i s analys is , t h e t o t a l e f f e c t would
n o t be m a t e r i a l l y d i f f e r e n t .
This i s due t o the h i g h l y i d e a l i z e d nature o f t h e present
For instance, i t i s i d e a l i z e d here t h a t the shock t r a n s i t i o n takes
I n a c t u a l i t y , a succession o f waves appears i n the f low, b u t
This hypothesis does n o t seem t o hold, however, when Mach d i scs
appear a t h ighe r Mach numbers.
does n o t possess the geometr ical s i m i l a r i t y c h a r a c t e r i s t i c s o f shock l a y e r
shapes.
by t h e i r h i g h l y unstable c h a r a c t e r i s t i c s , take the form o f a f l u c t u a t i n g
cone.
cou ld be extended t o the propagat ion c h a r a c t e r i s t i c s o f t r a n s i t i o n l aye rs , even
f o r h i g h e r Mach numbers, i n s p i t e of Mach d i s c appearance.
separate b a s i c i n v e s t i g a t i o n o f these phenomena seems desi r a b l e.
In t h i s case, t he successive shock format ion
On the contrary , the successive shock waves which are d i s t i n g u i s h e d
For t h i s reason i t i s feas ib le t h a t t h e i d e a l i z a t i o n o f c c n i c a l shapes
I n any case, a
- 57-
11. THE CONICAL SHOCK LAYER
I t was shown in the previous section t h a t the radiation character is t ics
of an a rb i t r a r i l y shaped shock layer were described by the forcing function:
( 1 1 . 1 )
w where k = - i s the wave number and the integral i s evaluated a t the surface
I) = I ) ~ . The subscripts i, j and k can take on any one of the values
between 1 and 3 , b u t i t i s self-evident t h a t i # j #k. This notation i s
more f lexible since i t allows the choice of any coordinate of the curvil inear
t r i p l e t as the shock surface.
aO i i
In the par t icular case of conical shocks i t i s convenient t o choose the
spherical coordinate system:
( 1 7 . 2 ) 3 J, = o 2 J1 = e 7
J, =;
2 so t h a t the conical surface i s represented by the coordinate I ) ~ = eo, where eo
i s the value of the conical ang e formed by the shock. Thus, referring t o the
notation in Equation ( 7 1 . 7 ) , i t i s inferred t h a t i = 2, j = 1 , and k = 3 .
I n view of the above defin t ions the shock region i s represented by
the coordinates :
Using the above notation, Equation ( 7 7 . 7 ) takes on the following form
- 58-
Let the observation point = [ ~ , n , c ] a l so be defined in terms of
spherical system of coordinates
5 = U i n y c o n v
n = U i n y n i n v
5 = Rcoay (77.5)
Under these conditions the quantity h = 1; - may be written
( 1 1 . 6 )
( 1 1 . 6 ~ )
I t i s now apparent t h a t i n the f a r f ie ld analysis the l inear dimensions
R from the origin t o of the shock layer are much smaller than the distance
the observation p o i n t . T h u s Equat ion ( 1 1 . 6 ) may be expanded in terms of the
r a t io ZIR
( 7 1 . 7 )
Consider next the integrand in Equation ( 7 1 . 4 ) . The quantity tr in the
denominator a f fec ts the magnitude of the radiated f i e ld b u t i t has no e f f ec t
on the phase of the emitted wave, which s modified by the wave number k .
Thus, in the denominator, l i t t l e e r ro r w 11 be affected by replacing 4 with R .
We wil l wri te Equat ion ( 1 1 . 4 ) in the following form:
and consider the expression in the bracket:
= -3- a e // e - d 2 F C O A o dF.4 ( 1 7 . 9 )
- 59-
which should be evaluated a t e = e o a f t e r the different ia t ion.
Relation ( 1 1 . 6 ~ ) f o r cosu th i s may be written as
Using
dF e d4 ( 7 7 . 7 0 ) - . ihFhiVl6dinyCVb ( 4 - V )
0 i' 0
L = a / e - i k F c o h e c o 6 y
a e
Employing the integral form of the Bessel function representation, the above
expression takes on the following form:
7 = 2 ? l a , a :ik:cvh e c a b y Jo ( k F b i n e h i n y ) d r ( 1 7 . 7 1 ) 0
Performing the different ia t ion and set t ing e = eo one gets
- i k F c a d e O c m y 7 = 2TiikbineOcvby e Jg ( k T h i n e o d i n y )Tdr t I' 0
Using these resul ts in Equation ( 1 1 . g ) the following expression for the
forcing function of the conical shock resu l t s :
- 60-
( 1 1 . 1 3 )
Equat ion ( 1 1 . 1 3 ) represents the forc ing f u n c t i o n o f a con ica l shock l a y e r
r a d i a t i n g acous t i c pressure waves i n t o the surrounding quiescent reg ion
sub jec t t o the s i m p l i f y i n g assumptions o f the present analogy.
t h e cone w i t h the z -ax is c f the coordinates i s represented by
l e n g t h o f the cone i s represented by the r a d i a l coord ina te such t h a t :
The angle o f
e = e,. The v
- ! J < p < L - -
where L i s t h e t o t a l l e n g t h o f t h e l aye r measured from the o r i g i n along the
con ica l surface. The diameter o f t he shock i s g iven by:
d = Gbxne0 thus P = 2Lbine0
P being the maximum shock diameter which i s assumed t o be o f t he o rde r o f
t h e nozz le diameter unless more exac t measurements a re ob ta inab le from
experiments.
Equation ( 1 1 . 1 3 ) i s p l o t t e d i n F ig . 6 and 7 showing constant i n t e n s i t y
l i n e s f o r a number o f f requencies.
I n add i t i on , s i m p l i f i e d computer runs f o r t he case o f parabo lo ida l
shock l a y e r s were performed us ing average cons tan t values f o r t he process
parameter B . These are shown i n Fig. 8. 2
The obta ined r e s u l t s i n d i c a t e t h a t , f o r low frequencies, t he d i r e c t i o n a l i t y
tends t o a l i g n i t s e l f w i t h the j e t axis, whereas the h ighe r frequency c o n t r i -
b u t i o n s swing away from t h e a x i s and tend t o be normal t o i t (e.g., shock
-61 -
l a y e r w i t h e o = 25'. see F igure 6) . A s i m i l a r t r e n d i s noted i n the case o f
eo = 4 5 ' , even though, i n t h i s case, the maximum h igh frequency d i r e c t i o n a l i t y
tends t o bu t does n o t reach 90" (see F igure 7). The same t rends h o l d t r u e
f o r other shock shapes (F igu re 8).
As mentioned prev ious ly ,* t h e o v e r a l l i n t e n s i t y o f r a d i a t i o n tends
t o a maximum i n the d i r e c t i o n s normal t o the shock l a y e r .
t h e re levant range o f Mach numbers the shock angles vary between 30' t o 60'
approximately (exc lud ing Mach d i s c s ) , i t i s t o be expected, i n the l i g h t o f
t he present theory, t h a t the maximum i n t e n s i t y w i l l tend t o occur i n the
same range w i t h respect t o t h e j e t ax i s .
Not ing t h a t f o r
* Section 10
-62-
y = 900
K=.2
y = I
y= m
Y = (
y = 9 o C
y = 90'
K = .5
y = i
y = 0'
Figure 6a.
f i e l d o f a 25" c o n i c a l shock showing t h e frequency dependent
P o l a r graphs o f l i n e s o f constarit i n t e n s i t y i n the t i o i s c
d i r e c t i o n a l i t y .
w h = - v a r i e s between .2 and 2 (cgs u n i t s ) aO
-63-
G f c P D
Q Z
Z
-I
2 v,
3 D z 0
L" D -I 2 1.2-
2
K = .2 SHOCK RADIUS = 2 COT 0s = 2.14
-
-
O S 2 t OO * 30 60
SPHERICAL ANGLE (DEGREES)
2.0 t 2 ' * O t
OO L 30 60
SPHERICAL ANGLE (DEGREES)
L
OO ' 30 60
SPHERICAL ANGLE (DEGREES)
G 2
a P
3
z L" D
z
z v)
Y
D z 0 z
1 . o t
60 0 0 30
SPHERICAL ANGLE (DEGREES)
Figure 6b. Distance from shock versus spherical angle f o r l ines
of constant in tens i ty i n the f i e l d o f a 25' conical
shock.
varies between .2 and 2 (cgs units} I2 = - w
"0
-64-
.
y =oo
y =I
1 Y =O'
Figure 70. Polar graphs of l ines of constant intensi ty in the noise
f i e l d o f a 45" conical shock showing the frequency
dependent d i rec t iona l i ty .
w k = - varies between .5 and 5 (cgs uni t s ) "0
-65-
K = .5 SHOCK RADIUS = 2.0 COTANGENToS = 1
n
z 1.2-
z
P z 0
-
2 v, -
0.8- -
0.4 - -
- 30 60 m 0,
0 SPHERICAL ANGLE (DEGREES)
K = 2 SHOCK RADIUS = 2 COT.eS = 1
0 0 1 30 60 9o
SPHERICAL ANGLE (DEGREES)
K = 1 SHOCK RADIUS COT.8,s = 1
OO I 30 60
SPHERICAL ANGLE (DEGREES)
I
K = 5 SHOCK RADIUS = 2 C 0 1 . 8 * ~ = 1
6.0 -
z
L m O O 30 60
SPHERICAL ANGLE (DEGREES)
Figure 7b . Distance from shock versus spherical angle fo r l ines of
constant in tens i ty i n the f i e l d of a 45" conical shock.
k = W varies between .5 and 5 (cgs un i t s ) . aO
-66-
N 0 1 s E
c 5 E c
0 0
S W E Q I C I I 4NGLE IDEGQEES)
M I C H NUNBERt 2 . 5 k 1 . t l
0 1 s I 4 N C E
c
0 Y
s b l 0 C I(
C N
DIST4NCE FqOM SL(0CK CM
4 C Q 0 5 s S v 0 C I(
-r E l
Q I O l I L OIST INCE I I O N C WtOC'C
Figure 8a.
i D i r e c t i o n a l i t y f i e l d of a P a r a b a l o i d a l Shock a t k = d a g = . lO(cgs u n i t s )
Upper l e f t g i v e s i n t e n s i t y v s . s p h e r i c a l angle . l i n e of c o n s t a n t i n t e n s i t y .
Lower l e f t shows a p o I a r p l o t of a
-67-
DIST4NCE FRO* WOCK CW
HOCK W4N - L m T u I N CY
F igu re 8b.
Directionality F i e l d o f a Paraba lo ida l Shock at k = w/a, = .50 (cgs units)
-68-
U C W NUUOERr 2.5 I= 1.m W O C R D l A K l E R = 4 A ¶ 1.m
0.00
1 1 0 4 .w
2.00 :: 1 0 0
'y Y O 0
8 C
L
C L
-2.00 8 0
0
7 0 -4 .m
60 4.m
-1CAL ANGLE (DECREES) Hoc6 %APE - LENGTH I N CW
-69-
,
0 1 s 1 A N C C
c (I 0 w s n 0 C I
C w
f
DISTANCL FRCU W E 6 CM Figure 8d.
Directionality Field o f a Parabaloidal Shock at k = ~ / a , = 2.00 (cgs u n i t s )
-?O-
12. THE TRIGGERING MECHANISM
The attempted t h e o r e t i c a l t rea tment o f non - i sen t rop i c f l o w
c h a r a c t e r i s t i c s makes i t feas ib le t o o f f e r some i n i t i a l specu la t ions con-
ce rn ing the causes and e f f e c t s o f the mechanism t r i g g e r i n g wave emission
f rom the entropy-producing regions.
d e r i v a t i o n , an entropy-producing reg ion i s governed by a non s e l f - a d j o i n t
(i .e. non-conservat ive) hyperbo l i c p a r t i a l d i f f e r e n t i a l equation. As a
consequence, an en t ropy wave i s propagated i n t o t h e d i s s i p a t i v e reg ion
whenever a p e r t u r b a t i o n appears on i t s boundary o r w i t h i n the reg ion i t s e l f
(non-homogeneous case).
assured by the second law o f thermodynamics which determines the d i r e c t i o n
o f the process.
d i s t o r t e d propagated wave p a t t e r n i s assumed, c a l l i n g f o r l i n e a r i z a t i o n
o f - and t h e van ish ing o f f i r s t - o r d e r d e r i v a t i v e s f rom - the equat ion
( s e l f - a d j o i n t form), i s e n t r o p i c f low cond i t ions are imp l i ed , f o r i n t h i s case
t h e propagat ion speed a = a ( S ) must be constant.
I n accordance w i t h the present
The s t i p u l a t e d growth o f the propagated wave i s
I n c i d e n t a l l y , i t may be r e a d i l y v e r i f i e d t h a t , when a non-
It i s now apparent t h a t , i n t h e l i g h t o f the present theory, the
behav io r o f such entropy-producing regions w i l l be determined by the
d i s s i p a t i v e e f f e c t s w i t h i n t h i s reg ion and the e x c i t a t i o n s imparted t o i t
a t t he boundary. I n the present approach of shock l a y e r a p p l i c a t i o n when
the th i ckness of t he reg ion tends t o zero the boundary e f f e c t s would tend
t o be predominant. As a consequence o f these e f f e c t s , t he re seems t o be a
-71-
marked d i s t i n c t i o n between shock l a y e r s (regarded here as entropy-producing
regions) formed by a supersonic mot ion o f a s o l i d i n an unbounded medium
and those formed i n the exhaust o f a supersonic nozzle.
case when almost un i fo rm c o n d i t i o n s p r e v a i l upstream and downstream o f t he
layers, t h e i r i n f i n i t e e x t e n t i n the remaining d i r e c t i o n s tends t o suppress
any poss ib le e x c i t a t i o n s f rom these sources.
formed i n the exhaust o f supersonic nozzles have t o contend w i t h t h e con-
d i t i o n s o f t he f r e e boundary a t t he t e r m i n a t i o n o f t he la-yer i n a d d i t i o n
t o i t s upstream and downstream per tu rba t i ons .
condi t ions p o t e n t i a l i n s t a b i l i t i e s a re formed i n the presence o f any
d i s s i p a t i v e e f f e c t s (e.g., shear l aye rs , v e l o c i t y f l u c t u a t i o n s , v o r t i c i t y ,
e t c . ) , b u t t he shock l a y e r s represent a f i r s t - o r d e r magnitude i n comparison
w i t h o t h e r sources, which i n t u r n enhance t h e i r i n s t a b i l i t y by appearing a t
t h e f r e e boundary where these shock l a y e r s terminate.
noted t h a t the same unstable c h a r a c t e r i s t i c s occur i n a moving r i g i d body
when boundary-layer shock i n t e r a c t i o n s take p lace, s ince the e f f e c t of r i g i d
boundaries a t t he e x t r e m i t i e s o f t he shock a re e f f e c t i v e l y counteracted by
the f l u c t u a t i n g medium.
For i n the f i r s t
On the o the r hand, shock reg ions
Thus, f o r supersonic-exhaust
I t should a l s o be
-72-
. I n the present engineer ing approach, the c h a r a c t e r i s t i c i n p u t o f t he
boundary e x c i t a t i o n s i s represented by a ser ies o f randomly d i s t r i b u t e d
impulses imparted t o the Shock l aye r .
impulses i s both space- and time-dependenty b u t t h e i r random c h a r a c t e r i s t i c
f os te rs the b e l i e f t h a t ergodic theory a p p l i c a t i o n would be a j u s t i f i c a t i o n
t o represent them i n terms o f any one o f these independent va r iab les . As
a consequence, a time-dependent behavior i s used in t he present approach.
The actual phys i ca l na€ure o f these
I n t u i t i v e l y one cou ld expect the Four ier t ransforms o f these impulse
J2 I U I I L L I ~ ~ ~ ~ t o reflect d i f f e r e n t p r o b a b i l i t y c h a r a c t e r i s t i c s f o r d i f f e r e n t
frequency bands i n t o which these impulses are decomposed.
feas ib le t h a t t he c h a r a c t e r i s t i c F o u r i e r t ransform should a l s o be a
s t a t i s t i c a l aggregate o f random impulse funct ions a c t i n g upon t h e shock
l a y e r .
be expressed as a frequency dependent p r o b a b i l i t y d i s t r i b u t i o n .
h e u r i s t i c cons iderat ions such a d i s t r i b u t i o n should tend t o zero f o r l a r g e
frequency values t o e f f e c t i v e l y ensure the existence. o f t he F o u r i e r I n t e g r a l
I t i s thus
Accord ing ly t h i s c h a r a c t e r i s t i c Four ier t ransform could h o p e f u l l y
From
I t i s f e l t t h a t a separate, more r igorous i n v e s t i g a t i o n o f these
i n i t i a l cond i t i ons i s needed i n f u t u r e developments of the sub jec t mat ter .
However, f o r the present a p p l i c a t i o n , a simple rep resen ta t i on o f the
c h a r a c t e r i s t i c impulse f u n c t i o n w i l l be employed.
d i s t r i b u t i o n has been assumed f o r
Thus, t he respec t i ve
O < w < w o
t o be equal t o unity.. This i m p l i e s t h a t the f u n c t i o n r ( w ) , i n terms o f
which t h e time-dependence of the fo rc ing f u n c t i o n has been w r i t t e n [Equation
( 7 0 . 4 ) ] i s equal t o u n i t y . Thereafter the t ransform i s chosen t o f a l l o f f
- 73-
as - as grows large. This choice effect ively prevents the divergence
of the Fourier integral for large values of frequency, even t h o u g h i t i s by
no means unique.
w
Based upon the above model, i t was found from the computational e f fo r t s
t h a t the overall behavior of the resul tant theoretical curves could be
correlated with experimental d a t a in a straightforward manner with a s ingle
exception. Namely,
results had t o be sh
t o the employment of
effects and the Dopp
n a l l cases the frequency spectrum of the theoretical
f ted by a constant factor .
the present acoustical analogy in which convective
e r s h i f t were n o t included.*
This could be expected due
Employment of the above model for the disturbances tr iggering non-
isentropic wave emission completes the overall theoretical representation
of t h e forcing function governing the acoustic propagation of shock layers
as derived in Section 10.
the resulting noise f i e ld were accomplished by the application of random
noise theory.** The choice of the Poisson dis t r ibut ion as a representative
probability function for the successive occurrences of the emission phenomena
was decided upon.** A typical CRT plot of spectrum fo r the i n i t i a l conditions
st ipulated above i s presented i n Figure 9. From the resul ts of the computation
i t can readily be seen t h a t empirical resu l t s also indicate the necessity of
the distribution t o tend t o zero for large values of frequency [see Figure 9
a n d Figure 1 1 ) .
( E . B. S m i t h , 1966) fo r Y = 50"
The computation of power spectral densit ies of
The plot represents a simulated run based upon Smith's Report
* Ribner, 1962.
** Rice, 1941, pp. 294-325.
- 74-
5 V
-7- I I I I I I I I I I I I
oI A
I I I I I I I 3 > 1 )
F igure 9. I n p u t Funct ions (S imula t ion of Smi th 's Experiment; Y = 50'.
A C h a r a c t e r i s t i c CRT Plot of Spectrum f o r Two Typ ica l Time
- 75-
0 3
13; EXPERIMENTAL VERIFICATION
Correlation of the obtained theoret ical resu l t s w i t h experiments was
attempted by actual simulation of given nozzle and flow charac te r i s t ics .
In certain cases conjecture had t o be used concerning the shock s t ructures
o f the nozzle under consideration, b u t whenever possible these were obtained
f r o m personal contacts .* Thus, actual simulation runs were carried out
using appropriate values f o r shock layer angle, i t s diameter, upstream Mach
number, e t c . The various flow parameters used i n the simulation o f Smi th ' s
experiments a re presented in Table 11.
and experimental correlations are shown i n Figures 10 t o 12.
The resul ts o f these theoretical
Two shortcomings of the present acoustical analogy became apparent
as soon as comparison w i t h the available data for a given simulation run
were made.
the theoretical computations was noted, i t s value being an absolute constant
for each nozzle simulated.
convective and Doppler s h i f t e f fec ts (Section 10) i t appears t h a t such
behavior o f the computed resu l t s could be avoided by taking these i n t o
account.
I n the f i r s t place, a shif t i n the frequency spectra of
Since the present analogy did n o t account fo r
Secondly, i t was found that the acoustic contribution o f the
theoretical model tends t o be overly exaggerated i n the neighborhood o f
the j e t ax i s . The reason fo r t h i s behavior may again be found i n the
* e.g., Martin Co. report , Courtesy E . 6. S m i t h .
-76-
s i m p l i f i e d cha rac te r o f t he present analogy, s ince the a t t e n u a t i v e
na tu re o f t h i s reg ion i s probably due t o the e f f e c t s o f r e f r a c t i o n
( R i bner , 1966) which were disregarded in the engi nee r i ng computations.
I
On the o t h e r hand, the o v e r a l l t h e o r e t i c a l r e s u ? t s and the c h a r a c t e r i s t i c
t rends of j e t no ise phenomena seem t o be c lose ly f o l l o w i n g t h e general
pattern cf experimental f ind ings . From the s i m u l a t i o n runs o f Smi th 's
r e p o r t t he f o l l o w i n g f a c t s may be noted.
a) I n bo th the theory and experiment the d i r e c t i v i t y o f h ighe r f r e -
quency c o n t r i b u t i o n s h i f t s toward 90° f rom the j e t a x i s (F igure 10).
The computed r e s u l t s show a d i r e c t i v i t y s h i f t back towards a 65"
ang e from the a x i s f o r frequency inc rease above 3200 cps
( F i u re 6b, K=2.0 p l o t ) .
i n d i c a t e an i d e n t i c a l t rend, a 2.5 db i n t e n s i t y drop occu r r i ng
between 70" and 90" from t h e ax i s f o r cen ter band frequency o f
10 cps. [Smith, page 8-3, Program F i r i n g #l].
The t o t a l r e l a t i v e t h e o r e t i c a l i n t e n s i t y change (as a f u n c t i o n o f
t h e spher i ca l angle Y) i s computed t o be o f t h e o rde r o f 12 db.
The above c h a r a c t e r i s t i c and the ac tua l dec ibe l count a re w e l l
subs tan t i a ted by t h e experimental f i n d i n g s (F igure 10).
b)
The experimental data o f t h e r e p o r t
4
c )
-77-
d) The experimental measurements o f d e n s i t y spect ra e x h i b i t a
s u f f i c i e n t number o f data p o i n t s t o h i g h l i g h t c h a r a c t e r i s t i c
f l u c t u a t i o n s o f t he curve i n the h i g h e r frequency reg ions.
computed values seem t o a n a l y t i c a l l y con f i rm t h i s behavior
i n d i c a t i n g i t t o be a c h a r a c t e r i s t i c Bessel f u n c t i o n v a r i a t i o n
(F igu re 11).*
The computed s p e c t r a l c h a r a c t e r i s t i c s show a r e l a t i v e increase
db/decade i n the low frequency ranges.
The
e )
The measured
values c o n f i r m the v a l i d i t y o f t he above r e s u l t
o f about 20
ex pe r i menta
(F igu re 11)
f ) The o v e r a l l d i r e c t i o n a l i t y c h a r a c t e r i s t i c s o f t he t h e o r e t i c a l
model show a marked c o r r e l a t i o n w i t h the experimental measurements
(F igu re 12).
A p o i n t o f experimental cont roversy a r i s e s when the changes o f
i n t e n s i t y w i t h frequency are considered as a f u n c t i o n o f t he spher i ca l angle
Y. I n Smi th 's r e p o r t the d i r e c t i o n a l i t y curves have the same bel l -shaped
character, t h e maximum i n t e n s i t y va ry ing w i t h the frequency.
hand, both the General E l e c t r i c r e p o r t (Lee, Smith, e t a l . 1961) and the
F-1 engine data** i n d i c a t e an exchange o f energy takes place i n the range
On the o t h e r
* I n c o r r e l a t i n g the s p e c t r a l d e n s i t i e s , a conversion from the 1/3 octave
band ana lys i s used i n experiments has been e f f e c t e d .
** Courtesy G. Wi lhold, Unsteady Aerodynamics Group, NASA, MSFC.
-78-
.
40" < - - Y x 90". Thus, as the frequency increases, ' t h e maximum i n t e n s i t y a t
about 40" decreases and t h a t about 90' increases, t he curve p i v o t i n g about
an almost s t a t i o n a r y maximum value.
model tend t o support t he l a t t e r b u t n o t the former t r e n d
The t h e o r e t i c a l r e s u l t s o f t he present
- 79-
t 5 L s oc n
+ 8 1 1 f = 50 CPS
0 10 20 30 40 50 60 70 80 90
ANGLE FROM EXHAUST AXIS (DEG)
-8
-12
0 10 20 30 40 50 60 70 80 90
ANGLE FROM EXHAUST AXIS (DEG)
f = 200 CPS
0 -
MARTIN CO EXP
0 10 20 30 40 50 60 70 80 90
ANGLE FROM EXHAUST AXIS (DEG)
-8
-1 2
o 1 0 20 23 40 50 M) 70 80 9o
ANGLE FROM EXHAUST AXIS (DEG)
Figure 10. Model a n d as Determined Experimental l y by Smi t h .
Comparison of Frequency Directivity Indices as Computed by
-80-
.
9 0
0 0 m I1 2.
-81 -
c, 0 n
-82-
h- a-
C C
h
d c n p
3 0
9 0
0 f\ 0 8 m
(ea) 13A31 lV1133dS
U W E .P ca n c, 0 W ul 0 2= I- t c, .I-
3 c aJ V
r"
L C C C L 0-
Q z , E o 0
v c , 7 s
-83-
In
Y LL U w Z 0 u - I-
Q, U w @z
\ \ ‘ \ I I
I \’ I
-84-
TABLE I 1
Smi til I s S i inul a t i on
Shnck Diairieter 0 P \ I
Shock angle 25"
4 cm
Distance froin Shock
R 3756 cm
E x i t Mach Number 3.5
Non-diitiensi orial cutoff .6
W V
zO frequeticy,
-85-
CONCLUDING REMARKS
The preceding i n v e s t i g a t i o n o f non- isent rop ic propagat ion phenomena
was mot iva ted by the d e s i r a b i l i t y t o represent i n an a n a l y t i c a l manner the
c o n t r i b u t i o n o f d i s s i p a t i v e reg ions t o aerodynamic no ise generat ion.
eng ineer ing a p p l i c a t i o n o f t h i s ana lys is i s r e f l e c t e d i n the concept o f an
EPN (extended p l u g nozz le) device.
producing reg ions (shock waves) i n o rder t o a t te i i i i a te noise generat ion o f
o f high-speed nozz le exhausts. I n t h e present s tudy a mathematical model i s
formulated and analyzed f o r some s p e c i f i c cases t o determine t h e no ise
genera t ion c h a r a c t e r i s t i c s o f supersonic nozzles.
e x i s t i n g t e s t data appear t o c o r r e l a t e wel l w i t h t h e t h e o r e t i c a l r e s u l t s .
An
I t s f u n c t i o n i s t o modi fy ent ropy-
For these s p e c i f i c cases
As i t i s i n most cases, the present approach i s based upon a r a t h e r
s imple b a s i c idea which may have a tendency t o become l o s t i n t h e techn i -
c a l i t i e s o f t h e d e r i v a t i o n .
dynamic v a r i a b l e s which would d e p i c t non- isen t rop ic pressure f l u c t u a t i o n s .
I t s essence i s t h e ch0ic.e o f independent thermo-
It may be r e a d i l y seen t h a t i f , f o r instance, the pressure and ent ropy
be chosen as independent v a r i a b l e s , a1 1 pressure grad ien ts become i s e n t r o p i c
and, as a consequence, so do pressure f l u c t u a t i o n s .
unwarranted r e s t r i c t i o n upon the physical phenomena tak i r ;g p l a c e i n ent ropy-
producing r e g i ons.
Th is f a c t seemed an
I t should a l s o be remarked t h a t the choice of the thermodynamic J-
f u n c t i o n was based upon phys ica l asoects o f t h e preceding a n a l y s i s coupled
w i t h dimensional cons iderat ions. The actual d e r i v a t i o n o f t h e J process
was obta ined, however, f rom an ana lvs is o f convected ent ropy changes i n
-87-
*
e q u i l i b r i u m flows.
here due t o t h e i n i t i a l stages of i t s development and a l s o t o a v o i d
excessive compl ica t ions o f t h e present ana lys is .
t h a t t h e scope o f such a d e r i v a t i o n warrants a d d i t i o n a l a n a l y t i c a l e f f o r t s
i n a s u b j e c t which may be i n c i d e n t a l t o the propagat ion c h a r a c t e r i s t i c s o f
a f i n i t e ampl i tude pressure wave.
T h i s l a t t e r a l t e r n a t e approach has n o t been presented
I t i s f e l t , moreover,
-88-
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I -
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1 NASA-Langley, 1967 - 23 CR-736 -91 -
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